To determine the prime-power factorization of [tex]\(40 s^2 q^2\)[/tex], we must systematically break down each component of the expression:
### Step 1: Prime Factorization of the Number (40)
First, we'll factorize the number 40 into its prime factors:
- 40 can be divided by 2:
[tex]\[ 40 \div 2 = 20 \][/tex]
- The quotient 20 can also be divided by 2:
[tex]\[ 20 \div 2 = 10 \][/tex]
- The quotient 10 can again be divided by 2:
[tex]\[ 10 \div 2 = 5 \][/tex]
- Finally, 5 is a prime number.
Thus, the prime factorization of 40 is:
[tex]\[ 40 = 2 \times 2 \times 2 \times 5 = 2^3 \times 5 \][/tex]
### Step 2: Including the Variables
The given expression [tex]\(40 s^2 q^2\)[/tex] also includes the variables [tex]\(s\)[/tex] and [tex]\(q\)[/tex] each raised to the power of 2.
### Step 3: Combining All Factors
Combining the prime factorization of 40 with the variables, we get:
[tex]\[ 40 s^2 q^2 = (2^3 \times 5) \times s^2 \times q^2 \][/tex]
So, the complete prime-power factorization of [tex]\(40 s^2 q^2\)[/tex] is:
[tex]\[ 2^3 \times 5 \times s^2 \times q^2 \][/tex]
### Step 4: Verification with Provided Options
Now, we need to compare this factorization to the given options:
- [tex]\( 2 \cdot 2 \cdot 2 \cdot 5 \cdot s^2 \cdot q^2 \)[/tex]
- [tex]\( 2 \cdot 4 \cdot 5 \cdot s^2 \cdot q^2 \)[/tex]
- [tex]\( 2 \cdot 20 \cdot s^2 \cdot q^2 \)[/tex]
- [tex]\( 4 \cdot 10 \cdot s^2 \cdot q^2 \)[/tex]
The correct factorization [tex]\(2^3 \times 5 \times s^2 \times q^2\)[/tex] matches:
[tex]\[ 2 \cdot 2 \cdot 2 \cdot 5 \cdot s^2 \cdot q^2 \][/tex]
Therefore, the correct choice is:
[tex]\[ 2 \cdot 2 \cdot 2 \cdot 5 \cdot s^2 \cdot q^2 \][/tex]
